Methods for calculating errors of a function whose arguments have individual errors.

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8 views

Error propagation in expression containing estimates of mean values

How can I estimate the propagated uncertainty in the following expression? $$ f(X_1,X_2)=\frac{\overline{X_1^2}-\overline{X_1}^2}{\bar{X_2}\overline{X_1}^2} $$ The uncertainties in the estimates $\...
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7 views

Error estimation from resampling

In my data analysis I must always keep the errors (or uncertainties). In physics, a result without an error estimate is worthless. Take this trivial example: I measure $x$ and convert it to the ...
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30 views

Propagation of uncertainty through a correlation with its own uncertainty

So I'm trying to do propagation of uncertainty for the first time (read: I'm a noob at this), and it's proving to be a challenge. I'm trying to estimate the uncertainty in the friction factor of the ...
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16 views

Find error on the inputs

Suppose that we have a model with an input and an output. The model is exact (no structural uncertainty). However there is an error in the output ( the error is detected from already given exact ...
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15 views

Uncertainty in a fractional count

What is the uncertainty (68% confidence level) of $N/M$, where $N$ is the number of entries that pass a cut and $M$ is the total number of entries? ($N$ and $M$ are both integers, and I'm interested ...
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10 views

Propagate errors in measured points to Simpson's numerical integral

I have a set of measured/observed $y(x)$ points, each with an assigned standard deviation: $$y: \{y_0\pm\sigma_{y_0}, y_2\pm\sigma_{y_2}, ..., y_N\pm\sigma_{y_N}\}$$ I use scipy's implementation of ...
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13 views

error propagation on the median

Suppose to have a set of exact data $x_i, i=1,\dots,N$ and to calculate its median value $m$. Then, a sound way to estimate the error $\delta m$ on $m$ would be bootstrapping. (I think...) But what ...
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19 views

Error propagation with correlation

I have two different multidimensional objects in two different conditions, therefore four different vectors of observations: $$V1_{cond1} = v1_{1,1},v1_{1,2},\dots , v1_{1,n}$$ $$V1_{cond2} = v1_{2,1},...
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50 views

Calculating uncertainties for histogram bins of experimental data with known measurement errors

I have a set of experimental data (with each data-point having its own measured uncertainty), and I wish to produce a histogram of it. The x values of the edges of each bin are already defined. The ...
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13 views

Error on the square root of a ratio of values

When I have two variables A and B for which the standard deviations are $\sigma_{A}$ and $\sigma_{B}$ respectively. I know ...
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12 views

How to mitigate the hierarchical error propagation in tree-structured classification

Suppose we have a multi-class classification problem, where the number of classes $K \geq 3$ We use a tree structure of multiple SVMs to divide and conquer the problem, with one example in the figure ...
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24 views

Estimate error of predicted value obtained by linear regression model

I have measured 2 parameters, r and p. Each parameter was measured in three technical replicates (n=3) per sample. r is measured directly. p is measured indirectly; the data obtained is output ...
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14 views

Error propagation: add errors in quadrature, or use a weighted standard deviation?

I have a measurement $x$ with a known uncertainty $\sigma_m$. I have a black box that can take an error-free measurement $x$ and produce a value $y$ with a known uncertainty $\sigma_{b}$ (which is ...
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55 views

Error in Linear Regression Parameters: Using mean measurement vs. all measurements

I have a set of measurements y taken at 17 different values of x, with 50 repeated measurements at each value of x. They follow a simple linear relationship y = mx + c, and I am fitting the parameters ...
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1answer
16 views

Combining measurements with known uncertainties

I have $N$ rulers that each have a different but known level of accuracy, e.g. one's a meterstick, one's a yard stick, etc. I measure the length of my table using each ruler. How do I combine ...
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34 views

Why can't I recreate the same population of x, using linear-least-squares, for a known linear system, Ax=b?

I have a linear system of equations in the form Ax = b. \begin{equation} \begin{bmatrix} a_{11} \pm \sigma_{a_{11}} & a_{12} \pm \sigma_{a_{12}} & a_{13} \pm \sigma_{a_{13}} \\ a_{21} \pm ...
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29 views

Error equation for distance between points in spherical coordinates

(Question brought over from here, after asking here) Consider two points with spherical coordinates: $a=(r_1,\theta_1,\phi_1)$ and $b=(r_2,\theta_2,\phi_2)$. The cartesian coordinates of the points ...
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56 views

uncertainties from Monte Carlo simulation and error propagation are different

Inspired by this post Is Monte Carlo uncertainty estimation equivalent to analytical error propagation?, I try to check it myself using a simple function f=A/B, where A is 10 with uncertainty 1 and B ...
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2answers
136 views

Propagation of uncertainty through a linear system of equations, Ax=b, where A and b are correlated

I have a linear system of equations in the form Ax = b. The elements of A and b were experimentally determined and as such have some uncertainty. Within each row, A and b are correlated. Between rows, ...
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19 views

Confidence interval for the median - continued

I have a set of values $x_i\quad i=1,…,N$ of which I can calculate the median M. Each $x_i$ has an error $\delta x_i$. The $x_i$ values are the result of a maximum likelyhood estimation and their ...
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57 views

Is Monte Carlo uncertainty estimation equivalent to analytical error propagation?

If I have a deterministic, analytic model, $y=f(x)$, I can analytically calculate the uncertainty in $y$ from a known uncertainty in $x$, $\sigma$. Or I can do a Monte Carlo integration: sample from ...
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28 views

Generalised linear models error distribution (continuous response)

I'm a bit confused about what error distribution I should use for the generalised linear models that I am running. My response variable is litter decay rate (k) (continuous, which runs from -1.5 to +1....
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26 views

Propagating uncertainties using random forest out-of-bag accuracy estimates

Let's say I train a random forest on some data and get an out-of-bag accuracy estimate of 90%. I then predict a quantity using this trained forest. What should be the uncertainty I give to that ...
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1answer
57 views

How to combine two measurements of the same quantity with different confidences in order to obtain a single value and confidence

Back in the lab at university, we were taught to measure the quantity of interest some number of times (call this N), and then calculate the standard error. The underlying assumption here is that you ...
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32 views

Adding Up Margins of Error

I'd like to add data from Census.gov, but I don't know how to add up the Margin of Error. Example, I have the estimate number of renters in Congressional District 1, (203,941 +/- 4,892) and the same ...
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24 views

Propagation of uncertainty when both axes depend on same variables

I have a data set y as a function of t measured at a distance L. Both t and L have a known standard deviation (i.e. the relative error on t changes with t). What I am now trying to do is to define a ...
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69 views

Monte Carlo error propagation

Consider a set $X$ of $N$ iid random variables, each one with its own standard deviation: $$X: \{x_1\pm\sigma_1, x_2\pm\sigma_2, ..., x_N\pm\sigma_N\}$$ Say I have a "black box" numerical function ...
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2answers
33 views

Error propagation in a linear model

I am currently interested in learning more on error propagation. At the moment I am trying to find out how to calculate the uncertainty of a value that is obtained from a linear model. For the linear ...
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49 views

Propagation of error in matrix multiplication involving an inversion

Hi I have a system $a$ = $B^{-1}x$ where $a$ and $x$ are 9x1 vectors and $B$ is a 9x9 matrix to be inverted Each element in $B$ is a product of two values with their own uncertainties and the vectors ...
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26 views

Serial dilution error propagation

How does serial dilution minimise the occurrence of experimental error? Does it not increase/propagate imprecision?
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6 views

Theoretically is model of error as erroneous as the erro itself?

I'm teaching a response model [i_1,i_2,...,i_n] --> <-1,1> i_n is <-1,1> I have chosen recurrent neural network but this might have nothing to do with my question. I will compare model with ...
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45 views

Do Monte Carlo perturbations capture all the uncertainty in prediction?

I have a model $M$ that I use to predict a value $y = M(\vec x)$. I have known one-$\sigma$ error bars on each input $x_i \in \vec x$. I want to know the one-$\sigma$ error bar on my prediction $y$. ...
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32 views

Error propagation with dependent variables

I've posted this in physics but without much help that I can apply to my problem. Based on Microdosimetry theory, trying to figure out error propagation for a lot of quantities that are produced from ...
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58 views

How to propagate uncertainty into the prediction of a neural network?

I have inputs $x_1\ldots x_n$ that have known $1\sigma$ uncertainties $\epsilon_1 \ldots \epsilon_n$. I am using them to predict outputs $y_1 \ldots y_m$ on a trained neural network. How can I obtain ...
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38 views

Propagate uncertainty of a parameter through a function

Suppose I have a probability distribution (in fact I've got a nice case where that distribution is Gaussian) on a parameter value. e.g. the parameter $x$ has $\mu = 3$ and $\sigma^2 = 1$. Now suppose ...
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33 views

Measuring Prediction vs Actual Difference

To find error we usually measure our predicted y vs actual y. Are there any error terms that measure the error in y AND the error in x, given y? In the image below the line is doing a pretty good ...
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35 views

Error propagation of correlated data

I have a dataset of 1-hour averages of a variable measured throughout a year. The goal is to get the sum of that variable for the entire year. Each 1-hour average has an error value associated with ...
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32 views

Estimating errors from optimization? (Genetic algorithm or otherwise)

I have a vector of observations $\vec x_{\text{obs}}$ that have been measured with known uncertainties $\vec \sigma_{x}$. I have a model $f$ that takes parameters $\vec \theta$ and produces values $...
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14 views

How to calculate the estimation error of portfolio variance using propagation results?

I am trying to find a conservative approximation for the propagated estimation error of a investment portfolio's variance (comprising two assets), given we know the estimation error for the variance ...
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3answers
502 views

Variance of an average of random variables

This seems like it should be a pretty common problem. I have four estimates of fishing effort, each with its own variance. For subsequent calculations, I want the mean of the four estimates, and a ...
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1answer
59 views

Are the parameters of Non-linear regression independent of each other?

I'm propagating error in the parameters determined by the following growth function... $$ \hat{y} = ae^\frac{t}{b} + (1- a)e^\frac{t}{c} $$ Say I have another model that uses the parameters {a,b,c} ...
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235 views

Backpropagation with Cross-entropy Cost Function

I'm using the cross-entropy cost function for backpropagation in a neutral network as it is discussed here: http://neuralnetworksanddeeplearning.com/chap3.html#introducing_the_cross-...
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2answers
4k views

Cross-entropy cost function in neural network

I'm looking at the cross-entropy cost function found in this tutorial: $$ C = -\frac{1}{n} \sum x [y \ln a+(1−y)\ln(1−a)] $$ What exactly are we summing over? It is, of course, over $x$, but $y$ and ...
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1answer
53 views

Error propagation and relative error

Say I have measured the value A = 50 +- 2 and from this I am calculating the value: B(A) = A^2 From what I understand I calculate the new error using error propagation to be delta_B =...
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1answer
41 views

How to consider propagated measurement errors and “statistical errors”

I come from an Engineering background, and I am familiar with some basics of error treatment. However, discussing with a friend over some data he had to analyze, we couldn't quite figure out what to ...
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17 views

Representing error when there are multiple different, linked error sources

Suppose I make some measurements of variable $X_i$ in separate experiments, and each has some error. In a separate (single) experiment I measure $b$ which is independent of $X_i$ but shared between. ...
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199 views

What is the correct way to interpolate error?

If I have a 2-D data, say $y = f(x)$, with error in the dependent variable, $\delta y$ in this case, and I want to interpolate this data set to a coarser independent variable grid, $x$, what is the ...
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81 views

Adding numbers with asymmetric uncertainties

I need to add a series of numbers with asymmetric standard deviations, such as $$5_{-2}^{+1} \,+ \,3_{-3}^{+1}.$$ Although I know it's common to add the upper errors ($\sigma_{\scriptsize{+}}$) and ...
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23 views

Propagation of uncertainties in functions not continuously differentiable

According to the Guide to the Expression of Uncertainty in Measurement as published by the Bureau International des Poids et Mesures (BIPM), the combined standard uncertainty $u_c^2$ for a function $y ...
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1answer
97 views

Disadvantages of uncertainty in modeling

I am preparing a presentation, my work mainly concentrates on uncertainty and sensitivity analysis. I was wondering if I can convince my audience by the importance of studying uncertainty in modeling. ...